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56 GENERAL DEVELOPMENTS Crar, 2 EXERCISES 1. Prove that for all sets 4, B, and € it is not the case that ACB&BECECEA. 2 ProvothutilA=4XBthenA=0. 3. Prove thatif 4 X B > O then the (A XB) = 0. 4 Prove an analogue of Theorem 46 for the following definition of ordered pairs: isa Cin 4 XB such that (UC) N (uy) = fe, toy). $ 2.10 Summary ef Axiomos. For convenient subsequent reference we summarize here the six non-redundant axioms introduced in this chapter, The union axiom is omitted because it was shown in 82.6 that it follows from the axiom ol extensionality, thc pairing axiom, and the sum axiom. These six axioms suífice for all developments in Chapter 3, which is con- cerned with relations and functions. Axiom of Extensiohality: (Vitec4 ze B >A =B. Axiom Schema of Separation; GBlvatreBeve A &o(r)). Pairing Axiom: avec AG :=2Ve= . Sum Axiom: GOV) (IB)REB&BEA). Power Set Axiom: aB(VOCEBSCESA). Axiom of Requiariy: AOS AecA&(Vycr>yg AL CHAPTER 3 RELATIONS AND FUNCTIONS 8 3.1 Operations ou Binary Relations. Jn everyday contexts we frequently speak of relations which hold between two, or among several, “things. Thus we may say that Augustus stood in the relation of stepfather to Tiberius, or that the relation of betwcenness hoids among three poimts, When we reler to relations in ordinary contexts we Insist that Lhere be some jutuitive deseription of the sort of connection existing between items which staud in à given relation. Fortunately this vague idoa of intuitive con-. Hectedness may be dispensed with in formal contexts and a relation may “pe defined simply as a set of ordered paits. We shall in this chapter be concerned almost entirely with the thcory of binary relations, that is, relations which hold between two things. Morcover, as we shall sce, the “theory of w-place relations may be constructed within the theory of binary telations, Consequently we omit the modifying adjective “binary” in the formal definition.* Derinrrion 1. 4 isa relation o (Valze A > (Ate = (ya). Tt is interesting to note that this is our first definition of à one-place relation symbol since Definition 1 of Chapter 2, which characterized the property of being a set, A natural idea would be that subsequent definitions con- cerning relations must, like the definitions following the definition of sets, be largely conditional in form. Iowever, this is not the case, and ncarly all of what follows applies to arbilrary seta, not just those special sets which happen to be relations. = The subsumption of n-place (or n-ary) relations under this definition may be exemplified by considering ternary, that is, three-place relations. *Definitions and theororms xo mumbered anew in each chapter. A reference to à definition or theorem vithout explicit mention of a chapter is to a definition ow thcorem in the same chapter as the reference, 57 58 RELATIONS AND FUNCTIONS Cuap. 3 A set À is a ternary relation if and only if 4 is a relation and (Vote A > (Ty) (30) Cho) te = (gy, 2), wu). On the other hand, notice that not for every intuitive relation which occurs in set thcory is there a corresponding set of ordered paírs. For instance, there is no set cortesponding to the inclusion relation between sets. In von Neumann sct theory there is a proper class which is the inclusion relation between sets, but there is no proper class corresponding to the inclusion relation betwecen proper classes. We begin systematic developments with three simple thcorems, after first introducing the useful notation: 2 Ay. DzrIsIrION 2. 2Ágyo(gNCA. Turorem 1. O is a relation. pRoor, Immediate from the definition of relations, since the empty set has no members. Tanonnm 2. Risárelation &SCRS is a relation. rroor. Let x be an arbitrary element of 8. Then by hypothesis 2€ R, whence, also by our hypothesis, there is a y and a z such that u= (a. o Hence, according to Definition 1, S is a relation. QED. THrorrM 3. Rand 'S are relations — RNS, RUS and RS are relations. Use of the variables “Rº and “8º in the last two theorems is no formal innovation, for all capital italic letters are set variab! it is meant to be merely suggestive of the fact that we ave intuitively thinking about those sets which are relations, although the theorems hold for avbitrary sets. If E is a relation then the domain of R (in symbols: DK) is the set ol all things « such that, for some y, (x, y)€ R. Thus if BR; = 10,1), (2,3) then DR, = 10,2). The range of R (in symbols: 4%) is the set of all things 7 such that, for some x, (x,y) € &&. Tbus ar, = (1,8). Sec. 3.1 RELATIONS AND FUNCTIONS 59 “The range of a relation is also called the counterdomain or converse domain. “The ficid of a relation À (in symbols; &R) ís tho union of its domain and range. Tor example, º o SR, = [0,1,2,85. In the obvious formal developments connected with the threc concepts of domain, range, and field the only difficult problem is to show that the intuitively appropriate set exists, As usual the definitions thomselves are axiom-free, DeriniTioN 3. DA = [ui (I(zA 9. That DÁ is the appropriate set is confirmed by the following theorem. THroRaM 4. ze DAS (Ty)(sA 9). rroor, By virtue of thc axiom schema of separation €1) aBvoveBoze UVA & (Tx 4 9). Wo want to establish the equivalence obtained from (1) by omitting: [62] z€ UUA, Consequently, we need to show that (2) is implied by the assertion that there is a y such that 63) qAgy. The following chain of implications serves the purpose. By Definition 2 we have from (3): (mncA, whence by virtue of the definition of ordered pairs fia), leullca. Thus by 'heorem 55 of Chapter 2 feje UA, and by virtuc of Theorem 55 again ze UU A, Tt thus follows casily from (1) that (4) (IB)V o)(v e Be (la A 9). The remainder of the proof simply requires routine manipulation of definition by abstraction. Perhaps it ill be useful again to put in the 62 RELATIONS AND FUNCITONS Cuar. 3 By the routinc steps previously taken (see, e.g., proof of Theorem 4), we infer from (4) and Definition 6 that (5) ze do (AI = (2) km). Our theorem follows by a straightforward application of quantifier logic to (5). Q.E.D. The strategy of this proof, like others which appeal to the axiom schema, of separation for establishment of the existence of some sect, naturalty falls. into two parts. First, it must be decided what set already known to exist has the desired set as a subset. Ilcre thc answer is that the set À is in- tuitively a subset of the Cartesian product ol the range and the domain of À. Second, it must be proved that satisfaction of the condition q im the axiom of separation implies membership in the larger set. Ilere this consists of showing that (2) implies (8). When these tro parts of the proof are completed, it is usually a routine matter to finish the remainder, We now turn to some theorems about the converse operation; their intuitive content should be obvious. Tmronrey 12. Ás a relation, proor. Jf ze Ã, then by virtue of the definition of the converse operation and Theorem 47 of Chapter 2 Cy (Te) (e = (9,9); the theorem then follows from the definition ol relations. Q.ED. Trrorny 13. ÁCA. Turossm lá. Risarelation >R = R. Three distributive laws are next. a Terorem 15. ANB=ANB. proor. By virtue of Theorem 12 and Theorem 3 it is clear that wo need oniy consider ordered pairs; cm” TAÁNByogÁnBo oyArz&yBa eziyázBy co atânday. QE.D. (Note that in listing a sequence of equivalences we do not now justify each step when the inference is obvious from previous theorems,) Si Tmrorgm 16. AUB=4AUB. e Trrorem 7, A-B=4-B Smo. 3.1 RELATIONS AND FUNCTIONS 63 The notion which it is natural to introduce next is that of the relative product of two sets. Tf Zé and S are relations then the relative product of R and S (im symbols: R/8) is the relation which holds between x and y “3£ and only if there exists a z such that R holds between x and £, and S holds between z and y. Symbolically, wc have: Derrinrrion 7. 4/B = [(2,9): (do)(v Az &zBy)). T£, for instance, R 8 Ka, 3), (2,8) 63,0) then E/S = (1, 1), (2,1) S/B = 3,3). Proof of the next theorem, which uses the axiom schema of separation to settle the usual existence question, is left as an exercise, Terorem 18. c4/By— (Jo(vÃc&ez By). The proofs of the next four theorems are casy and will be left as exercises. “Tmzonzm 19. A/B isa relation. THzorsm 20. 0/4 = 0. Tarorem 21. D(A/B) CG DA. Tusorum 22. ACB&CCDS5A/CO B/D. Three thcorems asserting distributive laws follow; the proof of only one is given here, Turorem 23. 4/(BUC) = (A/BJU(A/0). proor. By virtue of Theorem 19 and the previously proved fact that --the union of two relations is a relation, we know at once that both 4/ (BUC) and (4/B)U(A/C) axe relations, Henee the following oquivalences establish our theorem: TAMBUO) yo (Jilrà 2 &e BUCY) (TA z&(eByv 20) SGivic&eByW)Vv (LÃzãzOy) CMzAz&eBW)V (Ev Az&zCy) sm A/ByV uv A/Cy en A/BIU(A/Oy. QED. 64 KELATIONS AND FUNCITONS Coap, 3 ec. 31 RELATIONS AND FUNCTIONS 65 Tazorey 30, RI(ANB) = (RIA) N(RIB). Timorem 81. R(AUB) = (RIAJU(RI|B). Tesonmy 32. RA mB) = (Bld) = (RB). Tarorem 33. (R/S)]A = (RIA)/S. The next definition introduces the notation*: RÁ, which is read: the image of the set A under R. Thusif E and 4 are defined as in the previous example 16 may be noticed that finding a procf of this sort depends upon some familiarity with the manipulation of quantific Tecorem 24, AMBNC) E (A/BIN(A/C). Turorem 25. (A/B) (A/C) CA/(B-€). The example following the definition of the relative product operation indicates that this operation is not commutative. Yhen combined with the converse operation, wc do get the following interehange of order: o Tunonrzyn 26. 4/B = B/4. RºA = (2,8), PROOF. ud Reto) = (IEdgar Guest). Darinrrion 9. RA = G(RIA). Most of the proofs of the thcorems concerning images of sets are omitted. To reduce the number of parentheses in the statement of these theorems, we uso the convention that “U?, “Nº, “=? dominate 'º. Thus A“AUB “is (RAJUB not RS(AUB). THroney 34, ve RAS (IleRybre 4). Tecorem 85. R(AUB) = RºAUR“B. rroor. yERGAUB) O(I)ERy&ze AUB) o (IulcRykrcA)v (GolaRy&se B) OyeERAV ye RB SyE R'AUKB, Q.ED, Terorwu 36. R(ANB)C RANRB, A 2 A/By o yA/Ba e (J)lyvA2z&zBa) o (ole Begcig) o TB/Ay QED. The next thcorem shows that the relative product operation is associative, and thus parentheses may be omitted without ambiguity in reiterated occurrences of tho relative product symbol. Tanorsa 27. (A/B)/C = AMBIO). The proof is omitted because it is a straightforward exercise in quantifier logic. . We now defino the notion of a relation's' domain being restricted to a given set. As usual the definition applies to arbitrary sets. Derisstion 8. RIA = RNA X G(B)).* “The definition may be illustrated by a simple example. Let, R = ((1,2), (2,3), (0, Edgar Guest) |, 4 =, A simple example shows that inclusion cannot be strengthened to identity in this theorem. Let R= 1,3, 0,3), Then 1— (01,8) (2,8) a . As, RA = 12), (2,8). 8 =p, Proofs of the following six theorems are left as exercises. Then Turonem 28. cRIAgorRy&ICA. Turorgm 29, ACB>RIAC IB, R/(ANB) —0, but REA NREB, = (8). “The notation RA follows that of Whitehead and Rus: The vertical lincis a standard notation for this notion. See, for instanco, Entatowsld : [1983, p. 12]. Crar. 3 68 RELATIONS AND FUNCTIONS (O) (VRREASRIREBSNAMACNA, (O) (VRRCAS R/É = > UA/UA = UA, (e (VIREM R/E=BSNA MS =nA? 2. Let R be the numerical relation such that aRyorty=1, Let A be the set of prime numbers between 10 and 20. Explicitly describe RIA. 22. Prove Thevrem 28. 23. Prove Theoremns 29-81. 24. Prove Theorems 32 and 83. 25. Let R be the numerical relation such that sRyoM+I=y. Let 4 be the set of integers. (a) What actis RSA? (b) What ses is RCA? (o) What set is (B/ICA? 26. Prove Theorem 34, 27. Prove Theorem 36, 28, Prove Theorems 38 and 39. 29, Prove that R$0 = 0, 5 3,2 Ordering Relations. Relations which order a set of objects oceur in all domains of mathematics and in many branches of the empirical selences. There is almost an endless number ef interesting Lheorems about, various ordering relations and their properties. Here we shall consider only certain of the more useful ones; a large number of additional theorems are included among the excreiscs, We begin with the fundamental properties of reflexivity, symmetry, transitivity and the like in terms of which we define different kinds of orderings. Because these notions arc so familiar, idustrative examples are given only spazingly.* Regarding generality of definition, the situation is the same as in the last section: the definitions hold for arbitrary sets, not for just relations. However, in order to inercaso the immediate intuitive content of theorems, in this section we shall systematically use letters 'R', “8º, and “Tº as set variables in those contexts in which the ideas being dealt with naturally refer to relations. But it should be strictly understood that use of the variables “Rº, “8”, and “7” does not entail any formal restriction on the definitions and theopems. For instance, we define the property of transi- *Examples and elomentary applications may be found among other places in Suppes a 1957, Chapter 10]. Sec. 3.2 RELATIONS AND FENCTIONS 69 tivity for arbitrary sets R, not merely for relations. Also, without intro- ducing a numbered definition we use henceforth the familiar notation; ves trvEA Eye A and pure A or LEA GUCA REC A eto. We begin with eight basic definitions. Risreflerive inà o (Vire A >SvRa). R is irveflcvive in AS (VALE A > (2 RO). Derinrrios 10, Derinrrion 11. DerinITION 12. R is symmetric in À VolvlaycA&eRy>gRe). R is asymneiric in A e (Volynlage A ksRy>-y Ra). R is antisymnetric in À e(VolvyluycAbaRytyhes =). R is transitive in À e(VolVnlvoleyz e A&eRy&yRa uk. DerFINITION 18. DerixirioN 14. DerrinrrION 15. KR is connected in A e(VoilvnleyelAtey>ozrRyvVyRo. E is strongly connected in A e (Volvilsyca>ecRyvyRa. In order to relate the above eight properties to the operations introduced in the previons section, it is more elegant to consider the corresponding one-place properties, that is, to deal with relations which are reflexivo, rather than reflexive in some set À, ete. The general definitions just given are useful later, For brevity we define the eight one-place properties with one fell swoop; the definitions are obvious: we simply take À to be the field of the relation. Derintrron 16. Derrintrion 17, reflexivo reflexive Deristrion 18. Rs : o Ris : im SR. sirongiy sirongiy connected connected In formulating the desired theorems, we need the notion of the identity relation on a set. Ft is clear from Theorem 50 of Chapter 2: that we cannot define an appropriate general identity relation, but what we can do is to define, for each set À, the identity relation 34 on A. (Thus “o RELATIONS AND FUNCTIONS Cuar. 3 the symbol '5' is not am individual constant designating the identity relation but a unary operation symbol.) Deraxtriox 19. 94 = (2,0): te A). Tn addition to the definition, for srorking purposes we need the usual theorem guarantosing that 94 is tho empty set only when we expect itto be. THroREM 42. rsÁgorcA. rroor. Since (0,0) = (0), (u,0)) = toi), it is clear that SAC OCA. Moreovex, it is casy to show that 45) «nc 4 tio) e ooA. By virtue of the axiom schema of scparation and the definition of abstrac- tion we may use (1) to obtain the thcorem. QED. In this and subsequent proofs which usc the axiom schema of separation to prove the existence of some set, we restriet ourselves to consideration of two crucial steps: deciding what set known to exist has the desired set as a subsct, and then showing that satisfaction of the appropriate condition + in the axiom (here is “x € 4') implics membership. in the larger set. 1 is perhaps elarifying to remark that the formal proof does not require the inference that SÁ CEA, although this easily follows. But the search for a set that has 44 as à subset is an essential strategic consideration in finding a valid proof. We state without proof three simple thcorems concerning identity relations. Tarornm 43. DIA=A. 'THrorEM 44. SA/9A = 34. Turogem 45. R isa relation O (9DR)/R = R. The eight theorems which follow could have been used as definitions, and they are often so preferred. In thc proofs familiar properties of the operations are used without explicit reference to the appropriate theorems. Tueonrax 46. R isreflevive> 5 RCA. Sec. 3.2 RELATION AND FUNCITONS a -prooF. [Necessity]. By Definition 17 cvery element in 9FR is of the form (e, , whence by Theorem 28, : € FR, and it then follows from the hypothesis that R is reflexive that (x, )€C R. [ouficieney) Let x be an arbitrary cloment of SR. Since our hypothosis is that SSRCR it follows at once that (ER, but then Risreflexive. QED. . This proof is fairly trivial, but it illustrates the approach to the remain- ing sevcn, most of which are not proved here. THzorEM 47. R is irreflevine +» RNgFR = 0. -R TazorEM 49. R is asymmetric o RNE = 0. TrronEM 48. R is symmetric es É Tmmorem 50, R is antisymmetric +» RNH E 9DR. Turoney 51. R is transitive o R/RCR. rRoor. [Necessity]. If 2R/Ry then there is a z such that cRzdeRy whence by the hypothesis of transitivity 2Ry. [Suíficiency]. From our hypothesis that R/RCR we have at once: (1) (Jo) (eRz&:Ry)>sRy, but it is a familiar fact of quantifier logic that (1) is logically equivalent to: (2) cRz&eRy-zRy. QED. THroneM 52. Ji is connected O (FR X FR) »9SR CR UÉ. proor. [Necessity]. (1) c[(SR X SR) = 95R]y then [29] sCTR&ge TRAS = y, 74 RELATIONS AND FUNCTIONS Car. 3 The sense in which a simple ordering or strict simple ordering is complete is expressed by the following theorem. Twrorem 61. RESCAXA&RandsS are strict simple orderings JASRE=S. We now want to introduce the important notion of a relation wcll- ordering a set. If R is a strict simple ordering of A then R wellorders 4 * every non-empty snbset of 4 has a first or minimal clement (under the relation 8). Actually, as we shall scc, we need assume only that R is connected in 4 rather than il is a strict simple ordering of À. “Fhe asym- metry and transitivity of & in 4 are then provable, as is the fact that any element of 4 except the last (under the relation &) has an immediate successor. Sinec this notion of well-ordering is somewhat subticr than the previous order notions introduced, the consideration of several examples will be a useful preliminary to the formal definition and theorems. In these ex- amples, as in previous ojes, we shall use integers, and in fact, real numbers, although thesc entities have not yct been defined formally within our system of set theory. Let N be the set of positive integers. Then N is well-ordered by tho relation less than, since each non-empty subset of N has a first element, namely, the smalfest integer in tho set. On the other hand, N is not vwell- ordered by greater than, since many subseis do not have fivst elements, in particular N itself. N does not have a first element with respect to > just because there is no largest integer. E The notion of a wellordering is so concéived that, unlike the other order properties considered so far, it is not invariant under the converse operation, that is, if R is a well-ordering, il does not follow that E is a wellordering. We already have one such example: < wellorders N, but < does not. A somewhat dificrent example is given by considering that is, nd. seas R A= a nisa positive integer PU (1). The set À is wellordered by <, but not by Z, that is, not by >. In this caso, the set À itself has a first element under the relation >, but the subsct A [1) does not. By appropriate modification of the definition of R-first clement of 4 we can frame the definition of well-orderings in such a fashion that either mi or < welhorders N, that is, we can let our well-orderings be simple Sc, 3.2 RELATIONS AND FUNCTIONS 7% orderings as well as strict simple orderings. To a large extent, the choice is arbitrary; we can, if we want, let a welkordering be neither of the two. For instance, if 4 = [1,2,3] and R =, i,(2, 2) (1,2) 2,3), (1,3)) then intuitively R well-orders A even though R is neither a simple ordering nor a strict simple ordering of 4. But this generalization is trivial, and there is one persuasive reason for choosing strict simple orderings rather than simple orderings: the membership relation is a strict simple ordering of the ordinal numbers as wc shall definc them, and, as we shall see, in Chapter 7, there is a natural connection between any well-ordering of à set and the well-ordering of the ordinals by the membership relation. We tum now to formal developments. T6 is technically convenient to distinguish between the notion of a minimal element and that of a first element. A minimal element has no predecessors, whereas a first element precedes every other element, Clearly, every fiest element is minimal, but not conversely. To prove asymmetry of well-orderings it is simpler to use the conecpt of a minimal element in their definition. DerINITION 26. x is an R-minimal element of AGLEA &(Vy) WE AS yRe). An obvious feature of this definition is that if RN(A X A) is empty then every member of A is an f-minimal element. Ilowever such degenerate situations are not of much interest; in the case of weil-ordorings, we get uniqueness of the minimal element. DerINITION 27, visan Rfirst elemento A vIcA&(VNVC A Gary ar). We next define «ell-orderings. Subsequently wo stato a simple necessary and sufficient condition in terms of asymmetry and first element in place of the concept of a minimal element. Drrinrron 28. R wellorders AR is connected in A & (VB) (BCA&B0>B has an R-minimal element). Wo now prove that under this definition K is asymmetric and transitive, THeorem 62. À wellorders A > R is asymmetric and transitive in À. rroor. To establish asymmetry, suppose by way of contradiction that there are elements x and in À such that z Ryandykz. Then, contrary to the hypothesis that & well-orders A, the non-empty subset (x,y) of A has no R-minimal element. For transitivity, suppose for some elements z,y2€C 4, wc havo a Ry and ykz,butnotxRz Since R is connected in 4, we must then have: 7% RELATIONS AND FUNCTIONS Crar, 3 zRº. Iowever, the subsct (2,72) does not then have an R-minimal element, for 2 R x rules out as the £-minimal element, « X y rules out y, and y Re rules out 2 Whence our supposition is absurd. QE.D, We leave as am exercise proof of the following three theorems. Turorem 63. R wellorders A & R is asymmetric and connected in À &(VB(BCA&B =0>B has an Rfrst element). Trrorem 64. R wellorders A&AH0O>A has a unique Rfirst element. Texorem 65. R wellhorders ALBC AR wellorders B. On the other hand it is, of course, not generally true that if R well-orders Aand STR, then S vell-orders A, Our next task is to prove the theorem about unique immediate successors. Two definitions are needed. Derixrrios 29. y.is an R-immediate successor of ao 2 Ry & (Vaz) (gRz>z=uV yRea. VerssirIoN 30. visar R-last celementof Axe A&(VlyeA &rHyoyRo). The definition of last clement is obviously similar to that of first element. in fact, we have: Trmorem 66. xisan R-last element of A — x is am ie-first element of A. The result concerning immediate successors can now be established. 'Irzorex 67. R wellorders A&ze Aa is not an R-last element of Az has a unique R-immediato successor. »roor. Consider B= [y: «Ry). By hypothesis the set B is not empty, since 2 is not the last element of the ordering, and it is easily seen that B has a unique first element, which is the immediate successor of à, Q.ED, In the theory of ordinal numbers it will be convenient to have the notion of an R-section and certain facts about such sections available. "The closely: related notion of the R-segment of » set generated by an element is also introduced. Derinirion 31. Bison R-sectionofA-BCA&A nÃ'BC B. "Thus a set B is an R-section of a set À if all R-predecessors in À of elements of B belong to B — obviously É“B is just the set of R-predecessors of » “elements of B. If, for instance, Sc. 3.2 RELATIONS AND FINCTIONS m : A=11,2,3,4] B,= 11,2) B,=0 B,= (2,8), then B, and Ba are <-sections of 4, but B; is not, since 1 < 2 and 1€ AB, On the other hand, B, is not a >-section oí 4, since 3 > 2 amd 3CA-B.. Derinrron 32, S(4,R,0)=|y: ycA&yRe). The notation: s(4,R,4) is read: the R-segment of A generated by q. The set S(A, R, x) is just the set of R-predecessors of 2 which are also members of 4. Trronem 68. CE A&R is transitivc in A>S(A,R,2º) is an R- section of A, proor. Supposey€ S(A, R, x). We need to show that tho R-predeces- sorg of y which are members of 4 are also members of S(4, R, x). Let 2 be such an R-predecessor of y, that is, ac AnÊSty), whence 1) 2Ry. Since ye s(A, R, 2), we have: (2) vBz, and thus by the hypothesis of transitivity it follows from (1) and (2); 2Rz, from which we conclude: 2zE S(A, R,2). QED, On the basis of this thcorem il is easily proved that Tyrorem 69. R wcil orders A > (Bisan R-sectionof A&BA ÃO re A&B = SA, R 2). Some further concepts of order such as those of an R-upper bound of x, an R-supremum of x, and a laitice are introduced in the exercises, EXERCISES 1. Prove the following: (9) Ri O Bis fo SA asymmctric— R is irrefloxive mmetric — R is antigymmetric symmetric and antisymmetrie 80 RELATIONS AND FUNCTIONS CHAP. 3 21. Prove Theorem 69. 22. What additional ordering hypotheses if any are needed to guarantee that if À and B are R-scctions of € then either 4C Bor BC A? 23. Consider the following definitions: (1) zis an R-lower bound of À (YpgyeA>zRy). (Gi) ais am Rinfimum of A e» «is an Rlower bound of À EV ty is an Rlower bound of Ay R a).* (ii) gs an R-upper bound 46 (Voce A=>2R 9). Gv) 3 is an R-supremum of 4 is an R-upper bound of À & (Va) (x is an R-upper bound of Ay Ra).t (v) A isa lattice relative to Re+ R is a partial ordering ofA& (VV A Eye A [uy has an R-supremum and an Reinfimum in À). (a) Construct two partial orderings of a set of five elements, one of which yiclds a luttice and onc of which does not, (b) How many distinct lattices can be construeted out of a set of threc elements? . (e) Prove thatif 4 isa Inttice relative to R then A is a lattice relative to KR. (d) Prove thatil X isa simple ordering of À then A is a lattice relative to R. (e) Give a counterexample to the assertion that if À is a Inttice relative to Rand BC A then Bisa lattice relative to R. () Prove that if À is a lattice relativo to Ry, and B js a lattice relative to R» then À X Bisa Inttice relative to tho relation À such that bt zu A and ye Bthen (om) Rue zRiukyReo. $ 3.3 Equivalenee Relations and Partítions. A relation which is roflexive, sypametrie, and transi + apuivalence relation on hat set. The most ubiquitous example i . The relation of parallelism betwcen straight lines is a familiar geometric example of an equivalenco relation; the relation of congruence between figures is another, The fundamental siguificance of equivalence relations is that they justify the application of a general principle of abstraction: objects which are equivalent in some respect generate identical classes. Analysis of equiva- lence classes of objects rather than of the objcets Lhemselves is often much simplor. The family of such equivalence classes of a given set 4 form a partition of the set, i.c., is a family of mutually exclusive, non-empty subsets of 4 whose union equals 4. Conversely, as we shall sce, every partition of a set defines a unique equivalenco relation on that set. For brevity wc define under the same number the appropriate one- and twwo-place predicates. DerintTION 83, $ 6 Risan equivalence relation o R is a relation & K is reflexive, 1 - symmetric, and transitive; E *An Reinfimum of 4 is often callod an A-greatost lower bound of À. +An A-supremum of A is also called an R-least upper bound of 4. Sxc, 3.3 RELATIONS AND FUNCTIONS 81 (ii) R is an equivalence relation on À «> À = GR &R is an equiva- tence relation. Unlike the ordering definitions in the last section, the present definition requires that be a relation. “Phe motivation for adding the additional requirement here is mainly terminological. “The phrase “K is am equivalence” is not desirable since “cquivalence' is used in several different senses in logie and set thcory. On the other hand, when the phrase “Z is an equiv- alence relation” is used, it scems odd not to require that 2 be a relation. A secondary motivation is provided by the simplicity of the next theorem. The demand in the definiens of (ii) thal 4 = SR is made for technical convenience in relating equivalence relations and partitions; the obvious- pess of this convenience will be apparent in the sequel. THroREM 70. R is un equivalence relation O R/R = R. The next theorem relates quasi-orderings and equivalence relations in a natural way. Tweorem 71. R is a quasiordering — ROR is an equivalence relation. The following definition introduces the notation: lx]; we call Rlz] the R-cosct of 2. Intuitively Rlx] is simply the set of all objects which stand im the relation R to x. When £ is an equivalence relation we also speak of R[2] as the R-equivalence class of x, Derinrrion 34. BlaJ= fyivRy). If F is the relation of fatherhood, i.e., « F y if and only if « is the father of 4 then FlGeorge VI] = (Elizabeth, Margaret) and FlThomas Aquinas] = 0, (Of couvse, F is not an equivalence relation.) As à simple artificial example, let R=81,1,0,2%, 6,3, (1,9, €, 1. Then R is an equivalence relation and Rn] = Bl] = 1,23, RlB] = (3). Note that in place of Definition 34, we could have uscd: Bla] = Ro). 82 RELATIONS AND FUNCTIONS Crue, 3 It is not customary in mathematics to be so explicit about the relation É by means of which thc equivalence class [+] is abstracted. However, it would be incompatible with our rules of definition to omit the frce variable “Rº in the definiendum. It needs to be emphasized that the notation: Rle] is non-standard and perhaps unique to this book, whereas the notation: fa] is freguently used. . We have the customary theorem whose proof depends on the axiom sehema of separation. Tunrorem 72. ye Ralo zrky. The following two thcorems put on a systematic basis the principle of abstraction mentioned at the beginning of the section. As we shall see, these two theorems provido the essential link between equivalence relations and partitions. Tumonam 73. zyCsR&R is an equivalence relation — (Rx) = Block. »roor. Assume: R[z] = Rlyl. Sinco R is reflexive, we have: yRy, and thus by the previous theorem ve By, whence by our assumption ve Rlal; and by virtue of the previous theorem again, x k y Assume now: «By. Let e be an arbitriry element of Rly]. Ia view of the previous theorem we have: vê a, whence, since R is transitivo, Ra, and thus 2€ Rial. We conclude that a) Rb] € Rlel. Now let « bo an arbitrary element of Rlxl; we have at once ZRau And since R is symmetric we have from our assumption: yRz, Sec. 3.3 RELATIONS AND FUNCTIONS sa whence by virtue of the transitivity of 2 yvRu, and : ue Rh. Thus (2) Rh) € Ely), “and we immedistely infer from (1) and (2) that Bl] = Ely. QED. The above proof illustrates a stratogy which is very common. We want to “:show that the sets Rx) and Rly] are identical. Et is not convenient to operate wilh a sequence of equivalences like those used in scveral previous “profs. Rather our strategy is to show that any arbitrary element of R[g] is a member of (2), and thus R[y] is a subset of Rfv]. Then we show that Rel is a subset of R[y]. These two results together establish the identity of thc two sets. “The second of the two theorems mentioned shows that equivalence - classes do not overlap. Tueorem 74. K is an equivalence relation —Rlt] = RlyV Ren Bly = 0. Noto that in this thcorem, unlike the preceding: one, there is no necd to Tequire that x and y bein SB, for if vg FA then Rlx] = 0, and the con- elusion of tho theorem is satisfied. We now tur to partitions. Roughly speaking, a partition of a set A is 2 family of mutually exclusive, non-empty subsets of 4 whose union “equals 4, For instance, if A=(1,2,3,4,5) and N = (1,2), 13,5), (4), then II is a partition of A, Formally, we havo: Derixrrios 35. IL is a partition of A e» UL = A&(VEXVOBEI &CEU&EBL0C>BNC=D&(Vo)ecIl> (Iy(ye 2). The use of the letter TI” has no formal significance, but refleets a practice that is both customary and suggestive, Notice that the last elansc of the definiens exeludes both individuals and the empty set from membership so RELATIONS AND FUNCTIONS Croar, 3 EXERCISES 1, Prove (1) (ENS) = Rlx] NS[x) (b) (BUS) = RIU SI]. Corresponding to (a) and (b) of Exercise 1, what holds for difference of sets? Prove Thcorem 70. Prove Theorem 71. Prove Theorem 72, Prove Thoorem 74, Let every member of 4 be an equivalence relation. Nos a to (a) Ts NA am equivalence relation? (b) Ts UA am equivalence relation? If so, give proof. 11 not, give counteroxample. 8. Give two partitions of the natural nurabors, one of which is finer than tho other. 9. Prove 'Theorem 76. 10. Prove Theorem 77. º 1). Prove that if R is à quasiordering then UML NÃ) is a partition of SK. 42. Prove Theorem 78. 13. Prove Theorems 79 and 80. 14, Prove Thoorem 81. 6 3.4 Functions. Since the cighteenth century, claxification and generalization of the concept of a function have attracted much attention. Fourier's representation of “arbitrary” functions (actually piecew ise con- tinuous ones) by trigonometric series encountered much opposition; and later when Weierstrass and Riemann gave examples of continuous functions without derivatives, mathemalticians refused to consider them seriously. Even today many textbooks of the diflerential and integral calculus do not give a malhemutically satisfactory definition of functions. An exact and completely general definition is immediate within our sct-theorctical framework. A funetion is simply a many-one relation, that is, a relation which to any element in its domain relates exactly one element in its range. (Of course, distinct clements in the domain may be related to the same element in the range.) The formal definition is obvious. Derintrion 39. fis a fúnction sf is a relation & (VV) afutefeogy =. The use of the variable *f' is not meant to have any formal significance. We use it here in place of “4” or “R to conform to ordinary mathematical usage. To summarize our usc of variables up to this point: ABC, RSS TD EG Suc. 3,4 RELATIONS AND FUNCITONS 87 are variables (with and without subscripts) which take sets as valucs; a. Dt to Bo E are variables (with and without subscripts) which take sets or individuals as values. In the case of functions we are not content to use the notation x /%, bui want also to have at hand the standard [unetional notation: f(x) = 4, where “f(x)' às read f of q! * Derrnirion 40. j(x) = yo Et) (efe) &afyv El) (vfo) & y=0. The definition is so frared that the notation (2)! has a definito meaning for any set f and any object 2. Tor example, if T= 1,0, 0,2) 6,4) then D=0 f2)=0 6) = 4. The operation of forming the composition of two functions is so exten- sively used in ecrtain branches of mathematics that various special symbols have been used for it; we usc a small cirelo “o”. Thus informally, Go gta) = Hgta). Composition is defined directly in terms of relative product; we introduce the new symbol 'o” rather than use the relativo product symbol because the order of and “7 in fo g' is the natural one [or functions and is the revorsc of that in the corresponding relative product term. Darintrion 41. fog =g/f. We have as two simple theorems: THronmm 82, f and q are functions —f Ng and fog are functions. TuzoREM 83. f and q are functions > (fo 9)(x) = fg(x)). Recalling the notion oÍ restricting the domain of a relation, we have: Turonzm 84, (fog) IA =fo (glA). Giranted that f is a function, we may strengthen two earlier theorems on 4he image operation (Thcorems 36 and 37). “Tn mathematical logic, following the usage of Whitehead and Russell in Principia Maihematica, tho notation: /ºz is olten used in place of: j(z). se RELATIONS AND FUNCTIONS Crar, 5 Turonem 85. fisa function >J(ANB) = =I(A MB). And we may strengthen the analoguc of Theorem 40 for the range of f. Thtouny 86. fis a function > (Q)NB = J(f“B). We also have: “Trronem 87. fesa function & ANB =0>fANFB = 0. We now define the notion of a 1-1 function. “A NFB EPA Fen Deriniriox 42. fisi-lesfand f ore functions. We have the obvious result: Tnrorem 88. fisll&r DEL EDS (fa) = flv) o 2 = To). When f is 1-1 a simple definition of its inverse is possible, Derinirion 48. fásl-lof = f Useful facts arc expressed in the following five thcorems. Tmrorem 89. fil iS( Gg) =2vorz =). Trzorem 90. fisll&ze DSI) = Tanonum 9. fisi-l&gye aff) = THeosEm 92. fandgarell>sfngisl-, Trmorem 93. fand gare l-l& SIND =0 & Rfnag = 0 >Sfugis Il. It is also desirable to define in a formal way at this point some standard mathematical language which wc shall use a great deal in later chapters. We summarize it in one definition, Derinrrion 44, () fisa function on (or from) A to (or ínto) Be f is a function EDf=A&RCB; Gi) fis a function from À onto Be fis a function & Df= A & af=B; (ii) f maps À into Bo fisa ll function & Dj=A&OCB; (iv) f maps A onto Besfisal-lfuncionk&Dj=A&A=B. The distinction between “into” and “onto” in this definition is standard in Lhe mathematical literature, and has its counterpart in ordinary usage: A 1-2 fune f maps À onto B when the range of f is the whole of B;. =it maps. 4, ánio. B when the rango of f is only some subset of B. Sec. 3.4 RELATTONS AND FUNCTTONS Eu We conctude this section by defining the set of all functions from 3 to 4, which is ordinarily designated: A?. This concept às useful in a wide varicty of mathematical contexts. = fifisafuncion&Df=BERIGA). By virtue of the axiom sehema of separation we may establish the usual theorem. Turorem M. fc AP ofisafuncion&D=B&ERICA, DEFINITION 45. We state without proof five elementary theorems. Tezorem 95. 4º= 0). Timorem 96. 4<0>0"=0. Turonim 97. 4? =064=0&B 0. Tezonrem 98. AB = (ey): ve A). Tmrorem 99 A4ACB=>A4CBº. EXERCISES 1. State and prove a necessary and sufficient condition for the union of two functions to be a function. 2. Prove Theorems 82 and 83. 3. Prove Theorem 84, Prove Theorems 85 and 86. Prove “Lheorem 87. Prove Theorems 89-93, Prove that if f is 1-] then: (a) JANB) =fºANHB, (b) FAB) = JA [B. 8. Given thatf and gare 1-1, consider the following assertions. If an asser- Noca “E ton is true prove it. If false, give a counterexample. (8) fUgis ia, 6) Jgist-t, (e) fogisih, (d) FUf-Hs IA, (e) ANB = 04 UglBis 8) ANB=OSfANHgB= 9. Prove Theorem 94. 10. Prove Lhcoreme 95-99. 11. Consider Exercise 23 of $3.2, in which lattices are defined, We want to develop an equivalent formulation in torms of operations. Let 4 be a lattice relative to R, and 2, 4 € 4. Then we define: 2 Nany = Rinfimum of f;y) 2U Amy = Resupremum of (eg). 
